Hi Liz, and welcome.
By the overlap option, do you mean that you intend covering the floor of the run with 1/2” galvanised mesh and then extending this to cover the ground round the perimeter of the run outside the walls? It might possibly be less time consuming to go for this option, but I don’t think it would be so secure in the long run, as in time the mesh might bend, the ground might shift a bit, and there will always be the chance that rats will be able to tunnel in between the top of the mesh and the bottom of the sides. And also you’re stuck forever with a border of mesh round the outside edges of the run, taking up space that can’t be used for anything else, and possibly becoming a trip hazard in time. If you can simply bend the mesh up about 20-30 cms at the edges and fix it to the insides, you will have a much neater and more secure solution. You can fix it quite easily and strongly to the sides if you get lots of short cable ties and simply thread them tightly through both pieces of mesh about 20-25cms apart, which will hold the mesh firmly in place and in shape. Or with two people on the job, you can join it by ‘sewing’ the mesh together with lengths of soft garden wire, passed back and forwards between you to make big ‘stitches’ , which is how I did mine and it’s been fine for 14 years now. Not difficult.
I think that avian flu is still with us this year at the moment - you might like to check the thread in the Chickens Health section - there’s a lot in the news about large numbers of wildfowl and swans having to be put down on nature reserves etc. So our flocks should still be under cover at this time. I’m glad you’re planning with this in mind as I’m sure it’ll be with us for the foreseeable future. See http://poultrykeeperforum.com/viewtopic.php?t=11398&start=30
Good luck with the new project, and do keep us in touch with how you get on.